Hedging and the Value of Waiting to Investa

Glenn Boyle* University of Otago Department of Finance and Quantitative Analysis PO Box 56 Dunedin New Zealand [email protected] 64-3-479-8039 (phone) 64-3-479-8193 (fax)

Graeme Guthrie Victoria University of Wellington

27 January 2003

a

For helpful comments, we are grateful to Glenn Kentwell and Peter MacKay. Any remaining errors and ambiguities are our responsibility.

Hedging and the Value of Waiting to Invest

Abstract We analyze the optimal hedging policy of a firm that has flexibility in the timing of investment, but faces a constraint on its ability to raise external funds. By reducing the risk that the firm's ability to finance investment will disappear, hedging restores its timing flexibility and thus raises the payoff threshold required to justify investment. The principal implication of this result is that hedging adds value not only by allowing investment to occur, as is the case in Froot et al (1993), but also by allowing investment to be delayed in circumstances when it might otherwise have to occur prematurely. This observation may help explain the empirical findings that investment rates do not differ between hedgers and non-hedgers, and that hedging propensities do not depend on measures of growth opportunities.

Keywords: hedging, financing constraints, investment timing flexibility

Hedging and the Value of Waiting to Invest 1.

Introduction Firms enhance shareholder wealth by making productive investments. As a result,

Froot, Scharfstein and Stein (1993) argue that hedging allows firms to avoid the underinvestment problem associated with imperfect capital markets. When external financing is costly, a shortfall in internal funds may force a reduction in investment, which, in some states of the world, may be sub-optimal. By ensuring that internal funds are sufficient to allow all profitable investment projects to proceed, hedging adds to firm value. If failure to hedge causes a firm to under-invest, then, all else equal, hedging firms can be expected to invest more, on average, than non-hedging firms. However, evidence obtained by empirical researchers provides little support for such a view. For example, after adjusting for size differences, Graham and Rogers (1999, Table 4) and Allayannis and Mozumdar (2000, Table 1) report little difference in the level of investment between hedgers and non-hedgers, while Geczy, Minton and Schrand (1997, Table III) find that non-hedgers invest more than hedgers. Of course, there are some obvious explanations for these findings. First, non-hedgers in the above researchers' samples may simply have better investment opportunities than hedgers, so the positive effect of hedging on investment is offset by the smaller number of profitable investments that are available. Second, the link between hedging and investment rates is not necessarily an equilibrium interpretation of Froot et al's (1993) model, since firms that hedge may do so precisely because they face greater financing constraints. In this case, the propensity to hedge is an endogenous response to the external environment, reflecting the need to maintain parity with less-constrained firms possessing similar investment opportunities. However, neither of these explanations is clearly consistent with the data: in the samples of Geczy et al (1997), Graham and Rogers (1999), and Allayannis and Mozumdar (2000), hedgers are larger, and have significantly higher research and development expenditure, Tobin's Q, and/or market-to-book ratios, than non-hedgers. Thus, hedgers in

1

these samples appear to have better investment opportunities and face weaker financial constraints than non-hedgers. In this paper, we develop a simple model of dynamic investment and hedging that extends the Froot et al (1993) framework in a straightforward manner and, in so doing, suggests an alternative explanation for the apparent lack of any systematic relationship between hedging and investment rates. Specifically, our model considers the role of hedging when the firm has flexibility in the timing of investment. In this situation, a firm subject to external financing restrictions faces the risk that its ability to finance investment may disappear in the future. Consequently, it has an incentive to invest whenever it has sufficient internal funds available, even though the optimal unconstrained policy would entail delay of investment. Hedging, therefore, allows the firm to improve the timing of investment; without hedging, the firm might have to rush into investment. In other words, hedging adds value not only because it allows investment to occur, as in the static environment of Froot et al, but also because it allows investment to be delayed. Thus, over any intermediate period of time, investment can be unrelated to hedging: although hedging allows the firm to undertake more investment, it also allows it to delay more investment. As well as Froot et al (1993), our work also builds on several other studies. Grossman and Vila (1992) examine the optimal trading strategy of an investor subject to a financing constraint, but do not explicitly consider the roles of hedging and investment timing flexibility. Mello and Parsons (2000) analyze the role of hedging in protecting the value of a project's abandonment option, but do not address the investment timing decision. Boyle and Guthrie (2003) investigate the implications of external financing restrictions for a firm's investment timing decision, but do not allow the firm to hedge. In the next section, we present a simple numerical example that intuitively captures the role of investment timing flexibility in determining the optimal hedge position. In section 3, we develop this example further by extending the investment timing model of McDonald and Siegel (1986) to incorporate financing constraints and hedging. We show that the optimal hedging policy protects not only the firm's ability to undertake investment , but also its ability

2

to delay investment. In section 4, we summarize our findings and consider the implications of these for interpretation of empirical studies of hedging behaviour.

2.

A simple example This simple numerical example compares the hedging decision of a firm with

flexibility in the timing of investment with that for a firm without such flexibility. Although this example is highly stylized, it illustrates the basic insight of our story in an easilyunderstood manner. A firm has the rights to a $100 project and it must finance this from internal funds. There are three dates; we ignore discounting. Currently (at date 0), the project value V is $98 and the firm's cash stock X is $105. Subsequent evolution follows simple binomial processes: V rises or falls by 5% while X is either $5 higher or $5 lower. All outcomes occur with equal probability.

V2= 108.05 X2 = 95 V1 = 102.9 X1 = 100 X0 = 105 V1 = 93.1 X1 = 110

V2 = 97.8 X2 = 105 V2 = 97.8 X2 = 105 V2 = 88.4 X2 = 115

Figure 1.

Evolution of project value V and cash stock X without hedging. At date 0, the project is worth $98 and is 5% higher or lower at subsequent dates. The firm's cash stock X is $105 at date 0 and is $5 higher or lower at subsequent dates. All changes occur with equal probability.

3

At date 0, the firm can hedge date 1 cashflow by holding h units of an instrument that offers payoffs which are perfectly negatively correlated with cash stock innovations. As a result, date 1 cashflow is given by

X1 = 105 + ε(1-h)

(1)

where the innovation ε = 5 or -5, each with probability 0.5. We consider two investment scenarios and calculate the date 0 hedging decision for each. In the first scenario, the investment option expires at date 1, so the firm cannot invest at date 2. This corresponds to the standard static investment decision where the firm must invest at a particular date (date 1) or not at all. In this case, the firm invests in the high-V state (V = 102.90) at date 1, but not in the low-V state (V = 93.10). Moreover, internal funds are sufficient in the former state to allow the desired investment to occur, so no hedging is required at date 0. In the second scenario, the investment option expires at date 2, so the firm can invest at either date 1 or date 2. This represents a simplification of the standard dynamic investment decision where the firm has flexibility in the timing of an investment whose future payoffs are uncertain. The optimal investment policy is to exercise the investment option at a particular date if and only if the payoff from doing so exceeds the value of waiting and retaining the option. To analyze this more complex problem, we begin by first ignoring the uncertainty about X. At date 1, the firm considers investment if V = 102.90, but not otherwise (in the latter case, not only does the firm not invest at date 1, but the option itself is worthless since investment can never be profitable at date 2 either). If it invests, it obtains a payoff equal to (102.90 - 100) = 2.90. However, if the firm chooses to wait until date 2, it has a 50% chance of obtaining a payoff equal to (108.05 - 100) = 8.05, which from the perspective of date 1 has an expected present value of 0.5(8.05) = 4.025. Since 4.025 > 2.90, the value of waiting and retaining the investment option exceeds the date 1 payoff from exercising, so the optimal investment policy is to wait until date 2. However, incorporating uncertainty about X yields a different outcome. From Figure 1, we see that the date 2 payoff of 8.05 is not achievable, since X = 95 in that state, i.e., the firm has

4

insufficient funds to pay the investment cost of 100. Thus, choosing to wait until date 2 guarantees a payoff of zero, either because investment is unprofitable (V < 100) or because it is impossible (X < 100). Consequently, the date 1 value of retaining the option is zero. Since 2.90 > 0, the firm chooses to invest at date 1. Although this decision is optimal given the financing constraint, firm value is lower than in the unconstrained case. From the perspective of date 0, the value of the investment option is 0.5(2.90) = 1.45 when X is uncertain, but is (0.5)(0.5)(8.05) = 2.0125 if there is no risk of being unable to finance the project at date 2. Thus, the financing constraint reduces firm value by 0.5625. In this situation, hedging at date 0 is valuable because it prevents the firm having to invest prematurely.1 Suppose we set h = 1 in equation (1).2 Then the joint evolution of V and X is given by V2= 108.05 X2 =100 V1 = 102.9 X1 = 105 X0 = 105 V1 = 93.1 X1 = 105

V2 = 97.8 X2 = 110 V2 = 97.8 X2 = 100 V2 = 88.4 X2 = 110

Figure 2.

1

Evolution of project value V and cash stock X when the hedge ratio h = 1. At date 0, the project is worth $98 and is 5% higher or lower at subsequent dates. The firm's cash stock X is $105 at date 0 and is $5 higher or lower at subsequent dates. All changes occur with equal

Note that the firm is not allowed to hedge at date 1. To do so would simply repeat the first scenario in which the firm knows that it faces a "now-or-never" investment decision. This setup approximates, within a 3-date world, the situation faced by a firm with a perpetual investment option.

2

h = 1 is the minimum hedge necessary to restore the investment option. Thus, if hedging is costly, it is also the maximum hedge. If hedging is costless, then the firm is indifferent between all h ≥ 1.

5

probability. In the diagram above, uncertainty about date 1 cash is perfectly hedged at date 0.

With hedging, the risk of losing the ability to finance the project at date 2 disappears, so the firm can delay investment at date 1 confident in the knowledge that sufficient funding will be available at date 2 should it wish to invest at that date. Hedging thus restores the value of the investment option by eliminating the need for premature investment. The principal lesson of this example is straightforward. When a firm has flexibility in investment timing, hedging effectively provides it with an option on an option. That is, hedging makes it possible for the firm, should it so choose, to retain its option to delay investment. Without hedging, the firm may have to acclerate investment in a sub-optimal fashion and thus give up its option; hedging allows it to retain flexibility. The dynamic nature of the investment timing decision thus introduces an additional element into the hedging decision. At each date where it does not invest, the firm with timing flexibility knows that the optimal decision at the next date may be to invest, or it may be to delay further. Optimal hedging therefore requires that both alternatives be protected, i.e., hedging not only protects a firm's investment projects, but also the options on those projects. By contrast, optimal hedging in a static world of "now-or-never" investment decisions is concerned only with the former. Although this example provides valuable intuition, further insight into the joint determination of optimal investment timing and hedging requires a more complex model. We turn to this task in the next section.

3. 3.1

Hedging and investment timing flexibility The Model As in McDonald and Siegel (1986), a firm owns the perpetual rights to an investment

project and has the option to invest in this project at any time. If the firm invests, it pays a fixed amount I and receives a project worth V. Project value follows the geometric Brownian motion process

6

dV = µV dt + σV dε

(2)

where µ and σ are constant parameters and dε is the increment of a Wiener process. We assume that the project cannot be undertaken by any other firm. Thus, at each date, the firm can either exercise the rights and invest, or delay investment and retain the rights. In contrast to the McDonald and Siegel (1986) model, where the firm is implicitly assumed to have unlimited costless access to capital markets, we assume that the firm is restricted to financing the project with internal funds.3 More specifically, at the date the firm wishes to exercise its investment option, it can do so if and only if it has internal funds greater than or equal to I.4 Prior to that date, however, it can enter into hedging contracts that alter the future distribution of the internal funds. To model this constraint, we assume that the firm begins with an initial cash balance X which, over time, is augmented in three ways. First, if the firm does not launch the project, X is invested in riskless securities yielding the return r. Second, the firm has some existing physical assets with perpetual life that generate uncertain operational cashflow with dynamics

ν dt + φ dζ

(3)

where ν and φ are constant parameters, and ζ is a Wiener process with dε dζ = ρ dt. The third and final way in which the firm's cash stock changes over time is as a result of its hedging of operating cashflow. To model this activity in as general a manner as possible, avoiding the need to specify a particular type of hedging instrument, we suppose that there exists a traded asset (or trading strategy) whose returns are perfectly correlated with operating cashflow. The price x of this asset has dynamics

3

It is straightforward to allow the firm limited access to external funding (see, for example, Boyle and Guthrie, 2003), but as this has no qualitative effect on our results, we maintain the simpler structure here.

4

The source of this constraint is immaterial for our purposes, but it could be due to severe information or agency problems, or because the firm does not wish to reveal information to competitors about the project at the investment stage.

7

dx = µxx dt + σxx dζ

where µx and σx are constant parameters. At each instant in time, the value hx of the firm's hedging position must be maintained in a margin account earning interest at a rate ^r. This ensures that the firm cannot relax its financing constraint by simply selling the spanning asset; that is, shorting asset x can only be used to set up an ex ante hedge on operating cashflows, not to raise cash directly. Moreover, we assume ^r < r, so that hedging is costly. Finally, to ensure that the firm's positions in asset x constitute hedging activity, we require h ≥ 0.5 Thus, over the time interval dt, the firm's beginning cash stock yields interest rX dt, the margin account pays interest ^rhx dt, and the firm's existing assets generate cashflow equal to ν dt + φ dζ. Finally, the firm must inject cash equal to h dx into the margin account in order to maintain the required balance. Combining these sources of cash, the dynamics of the change in the firm's total cash stock are given by

dX = rX dt + ^rhx dt + (ν dt + φ dζ) − h dx

= (rX + ν + (^r - µx)hx) dt + (φ - σxhx) dζ

(4)

Let F(X, V) denote the value of the project rights for an arbitrary hedging strategy. This function satisfies the partial differential equation (see the appendix for details)

rF =

1 2 2 1 2 + 2 σ V FVV + 2 (φ - σxhx) FXX ρσ(φ - σxhx)V FXV + (r -δ)VFV + [r(X + G) - (r - ^r )hx] FX - (r - ^r )hx

where δ is the project's cashflow yield, G is the present value of the future cashflow generated by the firm's existing assets (i.e., G is a claim on the process in (3)), and subscripts on F

5

If h < 0, then the firm would not be hedging, but would instead be using the margin account to obtain a cheap loan for investment in asset x.

8

denote partial derivatives. Intuitively, the left-side of this equation is the expected return required by the firm in order to hold the project rights.6 The right-side is the expected return obtainable from holding the rights, given some hedging strategy h, i.e., the expected capital gain less the cost of hedging. The optimal hedging policy must therefore satisfy Bellman's equation rF = sup h≥0

{12

1 σ2V2 FVV + 2 (φ - σxhx) 2 FXX + ρσ(φ - σxhx)V FXV

+ (r -δ)VFV + [r(X + G) - (r - ^r )hx] FX - (r - ^r )hx}

3.2

(5)

A characterization of the optimal hedging policy Let h* denote the optimal hedging policy. From (5), the value of this position is given

by φ ρσV F (r - ^r) (1 + FX) h*x = σ + σ F XV + . x x XX σx2 FXX

(6)

Equation (6) does not provide a closed-form solution for the optimal hedging policy, since the functions FXV , FXX , and FX are all dependent on h*, but it does provide some useful intuition. The first two terms on the right side of (6) represent the optimal demand for hedging in the absence of transactions costs; the last term captures the reduction in this optimal demand due to the forgone interest (r - ^r) that hedging requires. To understand the meaning of the first two terms more fully, note first that any hedging policy typically reduces cashflow volatility by shifting cash from high-cash states to low-cash states. In equation (6), the first term gives the value of the complete hedge position, i.e., the hedge that completely eliminates random fluctuations in the firm's cashflow. However, shifting cash from a high-cash state to a low-cash state is counter-productive if the marginal value of cash is high in the former state, but low in the latter state. Consequently, the

6

Expectations are with respect to the martingale, or risk-neutral, probability measure.

9

optimal hedging policy does not shift cash from all high-cash states to all low-cash states, but only from high-cash states in which the marginal value of cash is low to low-cash states in which the marginal value of cash is high.

The second term in (6) reflects these

considerations. When the firm has investment timing flexibility, the marginal value of cash is given by FX.7 Thus, whether the optimal hedge is greater or less than the complete hedge position depends on the sign of ρFXV. If this term is positive, then fluctuations in V induce a negative correlation between the firm's unhedged cashflow and the marginal value of cash, i.e., cash tends to be high when it is most valuable. In this case, cashflow variability offers some natural protection against foregone investment opportunities, so the optimal hedge is less than the complete hedge. Conversely, if ρFXV is negative, then the marginal value of cash is negatively correlated with the firm's unhedged cashflow, so cash tends to be high in states where it is least valuable. In this case, the optimal hedge exceeds the complete hedge so as to obtain the desired correlation between cash and its marginal value. In neither case does the optimal hedging position fully insulate the firm from cashflow shocks. Instead, it seeks the degree of cashflow variability that best shifts cash from low-value states to high-value states. On the surface, this conclusion appears similar to that of Froot, Scharfstein and Stein (1993), but there is an important difference between their argument and ours. In their model, the direction in which the optimal hedge deviates from the complete hedge depends only on the sign of ρ, not on the sign of ρF XV . This difference occurs because of different assumptions about flexibility in investment timing. When investment cannot be delayed, additional cash permits the launching of projects that might otherwise have to have been abandoned entirely. In this case, the value of an additional dollar is always increasing in V; since more cash permits realization of the payoff (V - I), the value of additional cash is clearly higher when V is high than when it is low. Consequently, the sign of ρ is sufficient to determine the direction of the optimal deviation from the complete hedge. However, this need not be the case when investment can be delayed. Suppose, for example, that the firm is in a

7

It is straightforward to show that the hedging policy given by the first two terms in (6) minimizes the variance of FX.

10

situation where it would prefer not to invest immediately, but faces a high risk of funding shortfalls in the future. As a result, it has an incentive to invest prematurely. Extra cash in these circumstances reduces the risk of future funding difficulties, so the firm can afford to delay investment. But a call option value is less sensitive to the value of the underlying asset than is the exercise payoff, so FV is smaller following the cash injection than before it (when it would have exercised the option), i.e., FXV < 0. Thus, the value of an additional dollar can be decreasing in V when investment timing is flexible. Consequently, a firm with such φ flexibility may adopt a more-than-complete hedge (h*x > σ ) in circumstances where an x φ otherwise-identical firm without flexibility adopts a less-than-complete hedge (h*x < σ ), x and vice versa. More generally, the optimal hedging policy of a firm with investment timing flexibility will differ from that of a firm without this flexibility even when FXV > 0. When a project disappears if not taken at a particular date, the firm hedges to protect its ability to realise the project's launching payoff at the specified date. By contrast, if the timing of the project is flexible, then the firm wishes to protect not only the payoff from launching, but also the value of retaining the option to invest at a later date. The optimal hedge therefore has to consider both sources of value, not just the former. Put another way, when projects are of the "now-or-never" variety, cash has value because it allows the firm to invest, so insufficient hedging may cause the firm to forgo investment. Thus, hedging adds value because it allows investment to occur. By contrast, when there is flexibility in the timing of projects, cash also has value because it allows the firm to retain the option to invest, so insufficient hedging may cause the firm to invest prematurely. Hedging adds value because it allows investment to be delayed. The optimal hedge for a firm with timing flexibility can thus be greater or less than its inflexible counterpart. On the one hand, allowing the firm to choose the timing of its investment reduces its need for hedging since it does not lose the project if funding is not available on a given date. On the other hand, the need to protect the value of the investment option may increase the quantity of required hedging. For example, suppose X and V are perfectly positively correlated such that, at the next date, X exceeds I whenever V exceeds I. 11

Then the optimal hedge is zero if the firm must invest at the next date or not at all. But for firms with an ongoing option to invest, the value of this option is positive even when V < I and, moreover, this value is enhanced by additional X. Consequently, firm value would be increased by a hedge that moved cash from states where X is more than sufficient to finance investment to states where X is low. This way, the firm not only ensures that it has sufficient funds to invest if it wishes to do so, but also maximizes the value of its investment option should it choose to retain it.

3.3

Hedging and the optimal investment policy Further insight into the rationale for hedging can be obtained by examining its effect

on the optimal investment policy. For projects satisfying (5), the optimal investment rule is to launch if project value V exceeds some minimum threshold, but otherwise wait. When the firm is unconstrained, the threshold is a constant. However, because investment is allowed if and only if X ≥ I, the value of X places restrictions on the future states in which the investment option can be exercised, so the threshold is a function of X. Moreover, because stochastic fluctuations in future X can be altered by hedging, the threshold also depends on h. Thus, investment is justified if and only if

V ≥ V*(X; h)

where V* is the investment threshold function. Figure 3 illustrates the effect of hedging on the optimal investment policy by displaying V* as a function of X for different hedging policies.8 The bottom curve depicts the situation where the firm does not hedge. At low levels of X, the firm adopts a low threshold, reflecting the risk of future funding shortfalls. As this risk recedes with higher

8

For the purposes of this exercise, we use the following parameter values: I = 100, σ = 0.2, δ = 0.03, r = 0.03, ρ = 0.5, φ = 60, G = 100, σ x = ?, ^r = ?. With these values, we write the partial differential equation describing F(X, V) as a difference equation and solve this using the method of Successive Over Relaxation.

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levels of X, the threshold increases. When the firm follows the optimal hedging policy (top curve), the relationship between V* and X is similar, but V* lies everywhere above its unhedged counterpart. This reflects the role hedging has in reducing the risk of future funding shortfalls. By hedging, the firm can confidently delay investment in circumstances where it would otherwise have had to invest prematurely. In other words, hedging allows the firm to improve the timing of its investment, as well as the quantity.

V*

V* (optimal hedging) V* (no hedging)

X Figure 3. The investment threshold function: hedged and unhedged. The top curve depicts the value of the firm's investment threshold function when the optimal hedging policy is followed; the bottom curve corresponds to the case where there is no hedging. Hedging decreases the risk that the firm will have insufficient cash to finance the project in the future, thereby increasing the value of waiting and raising the investment threshold.

4. Concluding Remarks The fundamental goal of hedging for a firm with valuable investment opportunities that are available only at single future dates is to protect its ability to fund these projects as they become available. For projects that offer a choice of investment dates, however, hedging has an additional goal: to protect the ability of the firm to utilise this timing flexibility. Without hedging, the firm may have to rush into investment because of the risk that waiting exposes it to the risk of future funding shortfalls. By reducing this risk, hedging is valuable. Not only does it allow investment to occur, but it also allows investment to be delayed.

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This simple observation implies that the value of hedging depends not only on a firm's investment opportunities and financing constraints, but also on its investment timing flexibility. As a result, it has interesting implications for our interpretation of empirical work on hedging. First, although not subject to detailed analysis, existing evidence suggests that, after allowing for differences in investment opportunities and size, hedgers and non-hedgers have similar rates of investment, despite hedging making more investment possible. Our model suggests that this finding may be due to the ambiguous effect of hedging on the attractiveness of investment. On the one hand, hedging allows firms to undertake more investment by reducing the number of states in which there is a funding shortfall. On the other hand, it also reduces the risk of future funding shortfalls and thus makes waiting more attractive. Since these two effects work in opposite directions, it is perhaps not surprising that the data reveal little difference in investment rates between hedging and non-hedging firms. Second, empirical studies of the determinants of corporate hedging generally report little support for the view that more valuable investment opportunities enhance the propensity to hedge.9 Our model suggests that the absence of such a relationship need not be surprising, since the value generated by hedging a given set of investment projects depends on the timing flexibility offered by those projects. When timing is flexible, the optimal hedging strategy protects not only the projects themselves, but also the option to delay these projects until conditions are more favorable. With "now-or-never" projects, only the former consideration applies. Thus, firms with similar investment opportunities may adopt very different hedging policies, reflecting their different timing flexibilities.

9

See, for example, Nance, Smith and Smithson (1993), Berkman and Bradbury (1996), Mian (1996), and Graham and Rogers (1999). Two exceptions are Geczy et al (1997) and Gay and Nam (1998).

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References Allayannis, G. and A. Mozumdar, 2000. Cash flow, investment, and hedging. University of Virginia working paper. Berkman, H. and M. Bradbury, 1996. Empirical evidence on the corporate use of derivatives. Financial Management 25, 5-13. Boyle, G. and G. Guthrie, 2003. Investment, uncertainty and liquidity. Journal of Finance, forthcoming. Froot, K., D. Scharfstein and J. Stein, 1993. Risk management: Coordinating corporate investment and financing policies. Journal of Finance 48, 1629-1658. Géczy, C., B. Minton and C. Schrand, 1997. Why firms use currency derivatives. Journal of Finance 52, 1323-1355. Gay, G. and J. Nam, 1998. The underinvestment problem and corporate derivatives use. Financial management 27, 53-69. Graham, J. and D. Rogers, 1999. Is corporate hedging consistent with value maximization? An empirical analysis. Duke University working paper. Grossman, S. and J. Vila, 1992. Optimal dynamic trading with leverage constraints. Journal of Financial and Quantitative Analysis 27, 151-167. McDonald, R. and D. Siegel, 1986. The value of waiting to invest. Quarterly Journal of Economics 101, 707-727. Mello. A. and J. Parsons, 2000. Hedging and liquidity. Review of Financial Studies 13, 127153. Mian, S., 1996. Evidence on corporate hedging policy. Journal of Financial and Quantitative Analysis 31, 419-439. Nance, D., C. Smith and C. Smithson, 1993. On the determinants of corporate hedging. Journal of Finance 48, 267-284.

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Appendix Derivation of partial differential equation for F(X, V) The firm holds a portfolio consisting of a long position in the project rights F, a short position in asset x, and an interest-bearing margin account. Over the time interval dt, the change in the value of this portfolio is

dR = dF + ^rhx dt - h dx.

Using Itô's Lemma, this becomes

dR =

( 12

1 σ2V2 FVV + 2 (φ - σxhx) 2 FXX + ρσ(φ - σxhx)V FXV + µVFV +

[rX + ν + (^r - µx)hx] FX + (^r - µx)hx) dt +

((φ - σxhx)FX - σxhx) dζ + σVFV dε

Applying standard replication arguments yields

0

=

( 12

1 σ2V2 FVV + 2 (φ - σxhx) 2 FXX + ρσ(φ - σxhx)V FXV + φµx rφ (r - δ)VFV + [rX + ν - σ + σ + (^r - r)hx] FX + (^r - r)hx) dt x x

(A-1)

where δ is the project's cashflow yield. Further simplification can most readily be obtained if we assume the expected return µx is given by some equilibrium model such as the CAPM. In this case

µx = r + ρxmσxλ

where ρxm (= ρXm) is the correlation coefficient of the market return with dx, and λ is the market price of risk. Now let G denote the the market value of a claim to the future cashflow generated by the firm's existing physical assets. Since these assets are perpetual, dG = 0 over time interval dt. Thus, the return on a long position in G consists only of the current cashflow

1

(ν dt + φ dζ). Hence, a long position in G combined with a short position in φ/(xσx) units of asset x yields a total return of ν dt + φ dζ - (

φ xσx

x ) dx = (ν - φµ σx ) dt

Since this return is riskless, we must have φ x (ν - φµ ) = r ( G σx ) σx

which implies that φµx rφ ν - σ + σ = rG. x x

Making this substitution back into (A-1) yields the final form of the differential equation that F must satisfy 1 2 2 1 2 + 2 σ V FVV + 2 (φ - σxhx) FXX ρσ(φ - σxhx)V FXV + (r -δ)VFV + [r(X + G) - (r - ^r )hx] FX - (r - ^r )hx - rF = 0

2

||